Estimation of partially conditional average treatment effect by double kernel-covariate balancing

We study nonparametric estimation for the partially condi-tional average treatment effect, defined as the treatment effect function over an interested subset of confounders. We propose a double kernel weight-ing estimator where the weights aim to control the balancing error of any function of the confounders from a reproducing kernel Hilbert space af-ter kernel smoothing over the interested subset of variables. In addition, we present an augmented version of our estimator which can incorporate the estimation of outcome mean functions. Based on the representer theo-rem, gradient-based algorithms can be applied for solving the correspond-ing infinite-dimensional optimization problem. Asymptotic properties are studied without any smoothness assumptions for the propensity score func-tion or the need for data splitting, relaxing certain existing stringent as-sumptions. The numerical performance of the proposed estimator is demon-strated by a simulation study and an application to the effect of a mother's smoking on a baby's birth weight conditioned on the mother's age.